This is a part of a larger problem... but I am a little stuck on this little piece. I guess I am rusty on the stuff I learned last semester.
Problem:
$g$ is a monotonic increasing absolutely continuous function on $[a,b]$ with $g(a)=c, g(b)=d$.
$H=\{x : g'(x) \neq 0\}$
If $E$ is a measurable subset of $[c,d]$ then show that $F=g^{-1}(E)\cap H$ is measurable.
My thinking:
$g$ is monotone increasing as well as continuous, so it is a measurable function. Then $g^{-1}$ is also measurable? I know that this is not true in general. But I was thinking it is true for an absolutely continuous function. Then $g^{-1}(E)$ is measurable.
If $g$ is a measurable function then its derivative is measurable. $H=\{x : g'(x) \neq 0\} \implies H^c=\{x : g'(x) > 0\}\cup\{x : g'(x) < 0\}$ So, $H^c$ is a union of measurable sets, and therefore it is measurable. The complement of a measurable set is measurable, so then $H$ itself is measurable.
The intersection of two measurable sets is measurable, so the result follows.